The effect of profile step width on the determination of crystal structure parameters and estimated standard deviations by X-ray Rietveld analysis

Crystal-structure parameters and their associated estimated standard deviations (e.s.d.'s) have been determined for α-Al₂O₃ by Rietveld analysis of Cu Kα X-ray powder diffraction data collected on a standard diffractometer with profile step widths ranging from 0.01 to 0.32° 2θ. Estimates of the true precision of the parameters have been obtained by replication of the experiment five times. The degree of serial correlation between adjacent points in the profiles has been quantitatively determined by the use of the Durbin–Watson d statistic. Values of the conventional profile agreement indices, R_p, R_wp and goodness of fit, are not influenced by the scan-point density, but the Bragg R-factor decreases to a limiting value as the step width is decreased. The structural parameters obtained at all step widths are statistically indistinguishable at the 1σ level, although there is some loss of accuracy relative to single-crystal values at widths larger than 0.16°. The least-squares e.s.d.'s decrease linearly with the reciprocal square root of the number of steps in the pattern, as expected, but the e.s.d.'s of the unit-cell dimensions are the only ones that are significantly smaller than estimates of their true precision obtained from repetitions of the experiment. In the case of α-Al₂O₃, and possibly in general, the optimum step width for Rietveld analysis corresponds approximately to the minimum value of the full-width at half-maximum for well resolved peaks.